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The Quasicrystals Discovery as a Resonance of the Non-Euclidean Geometry Revolution: Historical and Philosophical Perspective

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Abstract

In this paper, we review the history of quasicrystals from their sensational discovery in 1982, initially “forbidden” by the rules of classical crystallography, to 2011 when Dan Shechtman was awarded the Nobel Prize in Chemistry. We then discuss the discovery of quasicrystals in philosophical terms of anomalies behavior that led to a paradigm shift as offered by philosopher and historian of science Thomas Kuhn in ‘The Structure of Scientific Revolutions’. This discovery, which found expression in the redefinition of the concept crystal from being periodically arranged to producing sharp peaks in the Bragg diffraction pattern, is analyzed according to the Kuhn Cycle. We relate the quasicrystal revolution to the non-Euclidean geometry revolution and argues that since “great minds think alike” there is a diffusion of ideas between scientific revolutions, or a resonance between different disciplines at different times. The story behind quasicrystals is an excellent example of a paradigm shift, demonstrating the nature of scientific discoveries and breakthroughs.

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References

  • Adelson, B. (2003). Issues in scientific creativity: insight, perseverance and personal technique. Journal of the Franklin Institute, 340, 163–189.

    Article  Google Scholar 

  • Alexander, E. R. (1984). After rationality, what? A review of responses to paradigm breakdown. Journal of the American Planning Association, 50(1), 62–69.

    Article  Google Scholar 

  • Axinte, E. (2011). Metallic glasses from “alchemy” to pure science: present and future of design, processing and applications of glassy metals. Materials and Design, 35, 518–556.

    Article  Google Scholar 

  • Bendersky, L. (1985). Quasicrystal with one-dimensional translational symmetry and a tenfold rotation axis. Physical Review Letters, 55, 1461–1465.

    Article  Google Scholar 

  • Bernal, J. D. (1958). The importance of symmetry in the solids and liquids. Acta Physica Academiae Scientarium Hungaricae, VIII, 269–276.

    Article  Google Scholar 

  • Bindi, L., Steinhardt, P. J., Yao, N., & Lu, P. J. (2009). Natural quasicrystals. Science, 324, 1306–1309.

    Article  Google Scholar 

  • Bindi, L., Steinhardt, P. J., Yao, N., & Lu, P. J. (2011). Icosahedrite, Al63Cu24Fe13, the first natural quasicrystal. American Mineralogist, 96, 928–931.

    Article  Google Scholar 

  • Blau, S. K. (2009). Natural quasicrystal found in a museum specimen. Physics Today, 62, 14.

    Article  Google Scholar 

  • Bogatyi, S. A. (2002). Metrically homogeneous spaces. Russian Mathematical Surveys, 57, 221–240.

    Article  Google Scholar 

  • Cahn, J.-W., Shechtman, D., & Gratias, D. (1986). Pauling’s model not universally accepted. Nature, 319, 102–103.

    Article  Google Scholar 

  • Chen, H., Li, D. X., & Kuo, K. H. (1988). New type of two-dimensional quasicrysal with twelvefold rotational symmetry. Physical Review Letters, 60, 1645–1650.

    Article  Google Scholar 

  • Clery, D. (2011). Once-ridiculed discovery redefined the term crystal. Science, 334, 165.

    Article  Google Scholar 

  • Connors, R. J. (1983). Composition studies and science. College English, 45, 1–20.

    Article  Google Scholar 

  • Conrad, M., Krumeich, F., & Harbrecht, B. (1998). A dodecagonal quasicrystalline chalcogenide. Angewandte Chemie International Edition, 37, 1383–1386.

    Article  Google Scholar 

  • Coppa, D. F. (1993). Chaos theory suggests a new paradigm for nursing science. Journal of Advanced Nursing, 18, 985–991.

    Article  Google Scholar 

  • Debnath, L. (2009). The legacy of Leonhard Euler - a tricentennial tribute. International Journal of Mathematical Education in Science and Technology, 40, 353–388.

    Article  Google Scholar 

  • Donmoyer, R. (2006). Take my paradigm … please! The legacy of Kuhn’s construct in educational research. International Journal of Qualitative Studies in Education, 19(1), 11–34.

    Article  Google Scholar 

  • Dotera, T. (2011). Quasicrystals in soft matter. Israel Journal of Chemistry, 51, 1197–1205.

    Article  Google Scholar 

  • Dotera, T. (2012). Toward the discovery of new soft quasicrystals: from a numerical study viewpoint. Journal of Polymer Science Part B, 50, 155–167.

    Article  Google Scholar 

  • Dubois, J. M. (2000). New prospects from potential applications of quasicrystalline materials. Materials Science and Engineering A, 294–296, 4–9.

    Article  Google Scholar 

  • Dubois, J. M. (2001). Quasicrystals. Journal of Physics: Condensed Matter, 13, 7753–7762.

    Google Scholar 

  • Dubois, J. M. (2008). Twenty-five years of quasicrystals: where are we now and what does the future hold? – A personal outlook. Philosophical Magazine, 88, 2351–2356.

    Article  Google Scholar 

  • Duparc, O. B. M. H. (2011). A review of some elements in the history of grain boundaries, centered on Georges Friedel, the coincident ‘site’ lattice and the twin index. Journal of Materials Science, 46, 4116–4134.

    Article  Google Scholar 

  • Elser, V. (1985). Comment on “quasicrystals: a new class of ordered structures”. Physical Review Letters, 54, 1730.

    Article  Google Scholar 

  • Elser, V., & Henley, C. L. (1985). Crystal and quasicrystal structures in Al-Mn-Si alloys. Physical Review Letters, 55, 2883–2886.

    Article  Google Scholar 

  • Farrugia, A. (2011). Falsification or paradigm shift? Toward a revision of the common sense of transfusion. Transfusion, 51, 216–224.

    Article  Google Scholar 

  • Frenkel, D. M., Henley, C. L., & Siggia, E. D. (1986). Topological constraints on quasicrystal transformations. Physical Review B, 34, 3649–3669.

    Article  Google Scholar 

  • Friedman, M. (2002). Kant, Kuhn and the rationality of science. Philosophy of Science, 69, 171–190.

    Article  Google Scholar 

  • Gilead, A. (2012). Shechtman’s Three Question Marks: possibility, impossibility, and quasicrystals. Foundations of Chemistry, 1–16.

  • Guo, J. Q., Abe, E., & Tsai, A. P. (2000). Stable icosahedral quasicrystals in binary Cd-Ca and Cd-Yb systems. Physical Review B, 62, R14605–R14608.

    Article  Google Scholar 

  • Hargittai, I. (1992). Fivefold symmetry. Singapore: World Scientific.

    Book  Google Scholar 

  • Hargittai, I. (2000). Candid science: Conversations with famous chemists. London: Imperial College Press.

    Book  Google Scholar 

  • Hargittai, I. (2010). Structures beyond crystals. Journal of Molecular Structure, 976, 81–86.

    Article  Google Scholar 

  • Hargittai, I. (2011a). “There is no such animal”—Lessons of a discovery. Structural Chemistry, 22, 745–748.

    Article  Google Scholar 

  • Hargittai, I. (2011b). Dan Shechtman’s quasicrystal discovery in perspective. Israel Journal of Chemistry, 51, 1144–1152.

    Article  Google Scholar 

  • Hargittai, I. (2011c). Stubbornness: ‘Impossible’ Matter. In: Drive and curiosity: What fuels the passion for science (pp. 155–172). Amherst: Prometheus Books.

  • Hargittai, I., & Hargittai, M. (2000). In our own image: Personal symmetry in discovery. New York: Kluwer/Plenum.

    Book  Google Scholar 

  • Hargittai, B., & Hargittai, I. (2012). Quasicrystal discovery—from NBS/NIST to Stockholm. Structural Chemistry, 23, 301–306.

    Article  Google Scholar 

  • Henley, C. L. (1985). Crystals and quasicrystals in the aluminum-transition metal system. Journal of Non-Crystalline Solids, 75, 91–96.

    Article  Google Scholar 

  • Janner, A. (2007). Personal reflections on the history of aperiodic crystals from early days to the state of the art. Philosophical Magazine, 87, 2601–2611.

    Article  Google Scholar 

  • Jeanjean, T., & Ramirez, C. (2009). Back to the origins of positive theories: a contribution to an analysis of paradigm changes in accounting research. Accounting in Europe, 6(1), 107–126.

    Article  Google Scholar 

  • Jeong, H. C., & Steinhardt, P. J. (1997). Constructing Penrose-like tilings from a single prototile and the implications for quasicrystals. Physical Review B, 55, 3520–3532.

    Article  Google Scholar 

  • Klassen, S. (2006). A theoretical framework for contextual science teaching. Interchange, 37, 31–62.

    Article  Google Scholar 

  • Kleiner, B., & Lott, J. (2008). Notes on Perelman’s papers. Geometry and Topology, 12, 2587–2855.

    Article  Google Scholar 

  • Kléman, M. (1990). Topology of the phase in aperiodic crystals. Journal de Physique France, 51, 2431–2447.

    Article  Google Scholar 

  • Kuhn, T. S. (1959). Energy conservation as an example of simultaneous discovery. In M. Clagett (Ed.), Critical Problems in the history of Science (pp. 321–356). Wisconsin: The University of Wisconsin Press.

    Google Scholar 

  • Kuhn, T. S. (1962a). The structure of scientific revolutions. Chicago: University of Chicago Press.

    Google Scholar 

  • Kuhn, T. S. (1962b). Historical structure of scientific discovery. Science, 136, 760–764.

    Article  Google Scholar 

  • Kuhn, T. S. (1970). The structure of scientific revolutions (2nd ed.). Chicago: University of Chicago Press.

    Google Scholar 

  • Lalena, J. N. (2006). From quartz to quasicrystals: probing nature’s geometric patterns in crystalline substances. Crystallography Reviews, 12, 125–180.

    Article  Google Scholar 

  • Levi, L., Rechtsman, M., Freedman, B., Schwartz, T., Manela, O., & Segev, M. (2011). Disorder-enhanced transport in photonic quasicrystals. Science, 332, 1541–1544.

    Article  Google Scholar 

  • Levine, D., & Steinhardt, P. J. (1984). Quasicrystals: a new class of ordered structures. Physical Review Letters, 53, 2477–2480.

    Article  Google Scholar 

  • Levine, D., Lubensky, T. C., Ostlund, S., Ramaswamy, S., Steinhardt, P. J., & Toner, J. (1985). Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Physical Review Letters, 54, 1520.

    Article  Google Scholar 

  • Lifshitz, R. (2011). Symmetry breaking and order in the age of quasicrystals. Israel Journal of Chemistry, 51, 1156–1167.

    Article  Google Scholar 

  • Lifshitz, R., & Diamant, H. (2007). Soft quasicrystals–Why are they stable? Philosophical Magazine, 87, 3021–3030.

    Article  Google Scholar 

  • Lilienfeld, D. A., Nastasi, M., Johnson, H. H., Ast, D. G., & Mayer, J. W. (1985). Amorphous-to-quasicrystalline transformation in the solid state. Physical Review Letters, 55, 1587–1592.

    Article  Google Scholar 

  • Lu, P. J., & Steinhardt, P. J. (2007). Decagonal and quasi-crystalline tilings in medieval Islamic architecture. Science, 315, 1106–1110.

    Article  Google Scholar 

  • Lubensky, T. C., & Ramaswamy, S. (1985). Hydrodynamics of icosahedral quasicrystals. Physical Review B, 32, 7444–7452.

    Article  Google Scholar 

  • Mackay, A. L. (1982). Crystallography and the Penrose pattern. Physica A, 114, 609–613.

    Article  Google Scholar 

  • Makovicky, E., & Makovicky, N. M. (2011). The first find of dodecagonal quasiperiodic tiling in historical Islamic architecture. Journal of Applied Crystallography, 44, 569–573.

    Article  Google Scholar 

  • Margenstern, M. (2010). Navigation in tilings of the hyperbolic plane and possible applications. Proceedings of IMECS (pp. 1–7), Hong Kong, 17–19 March.

  • Margulis, G. A. (1998). Aperiodic tilings of the hyperbolic plane by convex polygons. Israel Journal of Mathematics, 107, 319–325.

    Article  Google Scholar 

  • Martín, P., & Singerman, D. (2012). The geometry behind Galois’ final theorem. European Journal of Combinatorics, 33, 1619–1630.

    Article  Google Scholar 

  • Massimi, M. (2009). Philosophy and the sciences after Kant. Royal Institute of Philosophy Supplement, 65, 275–311.

    Article  Google Scholar 

  • Maxwell, N. (2012a). Arguing for wisdom in the university: an intellectual autobiography. Philosophia, 40, 663–704.

    Article  Google Scholar 

  • Maxwell, N. (2012b). In praise of natural philosophy: a revolution for thought and life. Philosophia, 40, 705–715.

    Article  Google Scholar 

  • Mermin, N. D., & Troian, S. M. (1985). Mean-field theory of quasicrystalline order. Review Letters, 54, 1524–1527.

    Article  Google Scholar 

  • Milnor, J. (1982). Hyperbolic geometry: the first 150 years. Bulletin of the American Mathematical Society, 6, 9–24.

    Article  Google Scholar 

  • Morgan, D. L. (2007). Paradigms lost and pragmatism regained –– Methodological implications of combining qualitative and quantitative methods. Journal of Mixed Methods Research, 1(1), 48–76.

    Article  Google Scholar 

  • Mosseri, R. (1992). Visible points in a lattice. Journal of Physics A: Mathematical and General, 25, L25–L29.

    Article  Google Scholar 

  • Pauling, L. (1985). So-called icosahedral and decagonal quasicrystals are twins of an 820-atom cubic crystal. Physical Review Letters, 58, 365–368.

    Article  Google Scholar 

  • Pauling, L. (1987). Evidence from x-ray and neutron powder diffraction patterns that the so-called icosahedral and decagonal quasicrystals of MnAl6 and other alloys are twinned cubic crystals. Proceedings of the National Academy of Sciences, USA, 84, 3951–3953.

    Article  Google Scholar 

  • Pauling, L. (1989). Interpretation of so-called icosahedral and decagonal quasicrystals of alloys howing apparent icosahedral symmetry elements as twins of an 820-atom cubic crystal. Computers & Mathematics with Applications, 17, 337–339.

    Article  Google Scholar 

  • Penrose, R. (1974). The role of aesthetics in pure and applied mathematical research. The Institute of Mathematics and its Applications Bulletin, 10, 266–271.

    Google Scholar 

  • Perelman, G. (2008). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159, 1–39.

  • Pihlströ, S., & Siitonen, A. (2005). The transcendental method and (post-)empiricist philosophy of science. Journal for General Philosophy of Science, 36, 81–106.

    Article  Google Scholar 

  • Ruse, M. (2005). The Darwinian revolution, as seen in 1979 and as seen twenty-five years later in 2004. Journal of the History of Biology, 38, 3–17.

    Article  Google Scholar 

  • Shechtman, D., & Blech, I. (1985). The microstructure of rapidly solidified Al6Mn. Metallurgical Transactions A, 16, 1005–1012.

    Article  Google Scholar 

  • Shechtman, D., Blech, I., Gratias, D., & Cahn, J. W. (1984). Metallic phase with long-range orientational order and no translational symmetry. Physical Review Letters, 53, 1951–1954.

    Article  Google Scholar 

  • Socolar, J. E. S., & Steinhardt, P. J. (1986). Quasicrystals. II. Unit-cell configurations. Physical Review B 34, 617–647.

    Google Scholar 

  • Socolar, J. E. S., Steinhardt, P. J., & Levine, D. (1985). Quasicrystals with arbitrary orientational symmetry. Physical Review B, 32, 5547–5550.

    Article  Google Scholar 

  • Steinhardt, P. J., & Bindi, L. (2011). Once upon a time in Kamchatka: the search for natural quasicrystals. Philosophical Magazine, 91, 2421–2421.

    Article  Google Scholar 

  • Steurer, W. (2004). Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals. Zeitschrift für Kristallographie, 219, 391–446.

    Article  Google Scholar 

  • Steurer, W. (2011). Quasicrystals: sections of hyperspace. Angewandte Chemie International Edition, 50, 10775–10778.

    Article  Google Scholar 

  • Steurer, W., & Deloudi, S. (2008). Fascinating quasicrystals. Acta Crystallographica. Section A, 64, 1–11.

    Article  Google Scholar 

  • Takakura, H., Gomez, C. P., Yamamoto, A., de Boissieu, M., & Tsai, A. P. (2007). Atomic structure of the binary icosahedral Yb–Cd quasicrystal. Nature Materials, 6, 58–63.

    Article  Google Scholar 

  • Tang, L. H., & Jaric, M. V. (1990). Equilibrium quasicrystal phase of a Penrose tiling model. Physical Review B, 41(7), 4524–4550.

    Article  Google Scholar 

  • Tennant, R. (2009). Medieval Islamic architecture, quasicrystals, and Penrose and Girih tiles: questions from the classroom. Symmetry: Culture and Science 2009 – Issue on Symmetry and Islamic Art, 1–8.

  • Tsai, A. P., & Yoshimura, M. (2001). Highly active quasicrystalline Al-Cu-Fe catalyst for steam reforming of methanol. Applied Catalysis A, 214, 237–241.

    Article  Google Scholar 

  • Tsai, A. P., Inoue, A., & Masumoto, T. (1987). A stable quasicrystal in Al-Cu-Fe system. Japanese Journal of Applied Physics, 26, L1505–L1507.

    Article  Google Scholar 

  • Tsai, A. P., Sato, A., Yamamoto, A., Inoue, A., & Masumoto, T. (1992). Stable one-dimensional quasi crystal in a Al-Cu-Fe-Mn system. Japanese Journal of Applied Physics, 31, L970–L973.

    Article  Google Scholar 

  • Vekilov, Y. K., & Chernikov, M. A. (2010). Quasicrystals. Physics - Uspekhi, 53, 537–560.

    Article  Google Scholar 

  • Wang, N., Chen, H., & Kuo, K. H. (1987). Two-dimensional quasicrystal with eightfold rotational symmetry. Physical Review Letters, 59, 1010–1017.

    Article  Google Scholar 

  • Williams, D. E. (2011). How concepts of self-regulation explain human knowledge. Winter, 16–21.

  • Wittmann, R., Urban, K., Schandl, M., & Hornbogen, E. (1991). Mechanical properties of single-quasicrystalline AlCuCoSi. Journal of Materials Research, 6, 1165–1168.

    Article  Google Scholar 

  • Yudin, V. V., Startzev, E. S., & Permyakova, I. G. (2011). The Fibonacci–Penrose semigroup formalism and morphogenetic synthesis of quasicrystal mosaics. Theoretical and Mathematical Physics, 167, 517–537.

    Article  Google Scholar 

  • Zeng, Z., Ungar, G., Liu, Y., Percec, V., Dulcey, A. E., & Hobbs, J. K. (2004). Supramolecular dendritic liquid quasicrystals. Nature, 428, 157–160.

    Article  Google Scholar 

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Acknowledgment

The authors would like to thank Istva’n Hargittai, Hungarian Academy of Sciences, Budapest University of Technology and Economics, for his constructive and detailed feedback on the manuscript. His personal correspondence and valuable advice contributed a lot to our understanding of the quasicrystals discovery.

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  1. School of Mechanical Engineering, Tel Aviv University, Ramat Aviv, 69978, Israel

    Dana Ashkenazi

  2. Department of Communication Systems Engineering, Ben-Gurion University, Beer Sheva, 84105, Israel

    Zvi Lotker

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  1. Dana Ashkenazi
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Ashkenazi, D., Lotker, Z. The Quasicrystals Discovery as a Resonance of the Non-Euclidean Geometry Revolution: Historical and Philosophical Perspective. Philosophia 42, 25–40 (2014). https://doi.org/10.1007/s11406-013-9504-8

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